Gas turbine engine airfoil shaped component

ABSTRACT

An apparatus comprises a gas turbine engine having a compressor. The compressor airfoil member is disposed in the compressor and has a camber angle variation along a chord and a concave pressure side surface at a tip portion of the compressor airfoil member from a leading edge to an intermediate-chord location. This first camber shape results in a normalized camber angle that reaches at least 0.4 in the first 10% of normalized chord, and the normalized camber angle is defined as 
     
       
         
           
             
               β 
               ′ 
             
             = 
             
               
                 
                   β 
                   i 
                 
                 - 
                 β 
               
               
                 
                   β 
                   i 
                 
                 - 
                 
                   β 
                   e 
                 
               
             
           
         
       
     
     where β i  is inlet metal angle, β e  is exit metal angle, and β is a camber angle at a given chord location.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a divisional of U.S. patent application Ser.No. 13/337,109 filed Dec. 24, 2011, which claims the benefit of priorityto U.S. Provisional Patent Application No. 61/427,720 filed Dec. 28,2010, both of which are incorporated herein by reference.

FIELD OF DISCLOSURE

The present invention generally relates to gas turbine engine airfoilshaped components such as vanes and rotatable blades, and moreparticularly, but not exclusively, to compressor blades and vanes.

BACKGROUND

Improving the performance of gas turbine engines remains an area ofinterest. Some existing systems have various shortcomings relative tocertain applications. Accordingly, there remains a need for furthercontributions in this area of technology.

SUMMARY

According to one aspect, an apparatus comprises a gas turbine enginehaving a compressor. The compressor airfoil member is disposed in thecompressor and has a camber angle variation along a chord and a concavepressure side surface at a tip portion of the compressor airfoil memberfrom a leading edge to an intermediate-chord location. This first cambershape results in a normalized camber angle that reaches at least 0.4 inthe first 10% of normalized chord, and the normalized camber angle isdefined as

$\beta^{\prime} = \frac{\beta_{i} - \beta}{\beta_{i} - \beta_{e}}$

where β_(i) is inlet metal angle, β_(e) is exit metal angle, and β is acamber angle at a given chord location.

According to another aspect, an apparatus comprises a gas turbine engineincluding a compressor for increasing a total pressure of a workingfluid prior to delivery to a combustor for mixing with fuel. Thecompressor has a plurality of airfoil members that extend between afirst flow surface and a second flow surface. The plurality of airfoilmembers have a first camber shape at a tip portion that provides acumulative integral according to the relationship

$0.4 < \frac{\int\limits_{0}^{1}{\left\lbrack {\int\limits_{0}^{x}{\beta^{\prime}{x}}} \right\rbrack {x}}}{\int\limits_{0}^{1}{\beta^{\prime}{x}}} \leq 0.5$

where x is chord location,

${\beta^{\prime} = \frac{\beta_{i} - \beta}{\beta_{i} - \beta_{e}}},$

β_(i) is inlet metal angle, β_(e) is exit metal angle, and β is a camberangle at a given chord location.

Other aspects and advantages will become apparent upon consideration ofthe following detailed description and the attached drawings whereinlike numerals designate like structures throughout the specification.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 depicts an embodiment of a gas turbine engine.

FIG. 2 depicts an embodiment of a compressor airflow member.

FIG. 3 depicts CFD analysis of an embodiment of a compressor airflowmember.

FIG. 4 depicts CFD analysis of an embodiment of a compressor airflowmember.

FIG. 5 depicts CFD analysis of an embodiment of a compressor airflowmember.

FIG. 6 depicts a camber angle.

FIG. 7 depicts a chart comparing an embodiment of an airflow memberagainst other designs.

FIG. 8 depicts a weighting distribution.

FIG. 9 depicts a performance of an embodiment of a compressor airflowmember.

DETAILED DESCRIPTION OF THE ILLUSTRATIVE EMBODIMENTS

For the purposes of promoting an understanding of the principles of theinvention, reference will now be made to the embodiments illustrated inthe drawings and specific language will be used to describe the same. Itwill nevertheless be understood that no limitation of the scope of theinvention is thereby intended. Any alterations and further modificationsin the described embodiments, and any further applications of theprinciples of the invention as described herein are contemplated aswould normally occur to one skilled in the art to which the inventionrelates.

With reference to FIG. 1, one embodiment of a gas turbine engine 50 isillustrated and includes a compressor 52, combustor 54, and turbine 56.In one form the gas turbine engine 50 can be coupled with an aircraft toprovide propulsive power. As used herein, the term “aircraft” includes,but is not limited to, helicopters, airplanes, unmanned space vehicles,fixed wing vehicles, variable wing vehicles, rotary wing vehicles,unmanned combat aerial vehicles, tailless aircraft, hover crafts, andother airborne and/or extraterrestrial (spacecraft) vehicles. Further,the present inventions are contemplated for utilization in otherapplications that may not be coupled with an aircraft such as, forexample, industrial applications, power generation, pumping sets, navalpropulsion, weapon systems, security systems, perimeter defense/securitysystems, and the like known to one of ordinary skill in the art.

In operation of the gas turbine engine 50 the compressor 52 receives aworking fluid such as air and compresses the working fluid before beingdelivered to the combustor 54. The combustor 54 mixes the compressedworking fluid with a fuel and burns the mixture to produce a core flowhaving products of combustion. The turbine 56 receives and expands thecore flow to produce power to drive the compressor 52. The gas turbineengine 50 can take a variety of forms including turbojet, turbofan,turboprop, and turboshaft. In addition, the gas turbine engine 50 can bea variable or adaptive cycle engine, can have centrifugal flowcomponents alternatively and/or additionally to axial flow components,and can include additional components than those depicted in theillustrative embodiment, among other possible variations. In short, thegas turbine engine 50 can take on a variety of different embodiments.

Turning now to FIG. 2, the compressor 52 of FIG. 1 includes an airflowmember 58 disposed within the flowpath and operable to alter a directionof a flow of working fluid passing through the compressor 52. In someembodiments the airflow member 58 can be a rotating compressor blade ora relatively fixed stator vane. The airflow member 58 can have anairfoil-like aerodynamic shape that includes characteristics such aschord, span, camber, mean camber line, and camber angle.

The shape of the airflow member 58 in the present application produces anumber of unique design-point and off-design-point characteristics ofefficiency, pressure ratio, and operating range. The shape also reducestip-gap flow, among other possible characteristics. As will be describedfurther below, the shape generally includes a relatively large change incamber angle for the first portion of the blade chord with a relativelyflat camber angle distribution for the remaining majority of the blade.To set forth one non-limiting example, the airflow member 58 can have arelatively large change in chamber angle near the leading edge thatoccurs within about the first 5% of blade chord. Near the leading edgeand tip of the airflow member 58 the pressure surface includes arelatively large concave region that serves to produce a relatively lowincrease in static pressure rise on the pressure surface. The relativelyflat camber angle distribution for the majority of the blade at the tipresults in a reduced suction-peak Mach number. The combination ofreduced suction-surface suction peak and the local static pressure onthe pressure-surface tends to reduce the overall driving pressuredifference across the rotor tip. The effect also serves to produce alocalized jet of stream-wise momentum flow. In addition to the shape ofthe camber angle just described near the leading-edge and tip of theairflow member 58, in some forms the airflow member 58 can also have arelatively large change in camber angle near its trailing edge.

FIGS. 3, 4, and 5 depict one form of the airflow member 58 and itseffects on static pressure and velocities. Computational fluid dynamics(CFD) analysis was performed on the airflow member 58 and shows areduced static pressure region, and resultant increased velocity region,near the leading edge of the tip region of the airflow member 58. Thedashed circle in FIG. 3 highlights the pressure surface of the airflowmember 58 in the tip region, and in particular highlights the localdecrease in static pressure relative to the rest of the airflow member58. FIG. 4, a view looking down the span of the airflow member 58, alsoshows the local decrease in static pressure of the pressure-side withinthe dashed circle. In addition, the view of FIG. 4 shows that thepressure-surface low pressure region occurs at approximately the sametrue chord fraction as the suction-surface pressure suction peak. Thepressure-surface static pressure is initially high near the leadingedge, it reduces along the chord of the airflow member 58 beforeincreasing again at approximately the 30%-40% true chord. FIG. 5 depictsrotor tip velocity contour. The dashed circle is again positioned nearthe leading edge of the airflow member 58 and it encircles a region ofthe pressure side where the relatively low pressure causes a localstream-wise acceleration of the flow which is roughly depicted by thearrow. The induced flow acts like a localized fluid jet. The increase instream-wise momentum, coupled with a reduction in the drivingcircumferential pressure gradient that induces the leakage flow acrossthe tip, reduces the magnitude and penetration of the leakage flow intothe flow passage. Such an effect can help to delay the onset of tipstall.

Turning now to a more detailed discussion of the properties of theairflow member 58, FIG. 6 depicts a camber line 60 extending from thepoint i, representing the inlet to the airfoil section of the airflowmember 58, to a point e representing the exit of the airfoil section.The points i and e are arranged along the line 62. In one form thecamber line 60 is the mean camber line and the line 62 is the chordline. The angle β_(i) represents the inlet metal camber angle and theangle β_(e) represents the exit metal camber angle. As will beappreciated, any point along the camber line 60 will have a camber angleβ just as the points i and e have a camber angle. In one form of theairflow member 58, the camber angle distribution of the airfoil sectionfollows according to the relationships

$\begin{matrix}{\beta^{\prime} = \frac{\beta_{i} - \beta}{\beta_{i} - \beta_{e}}} & (1) \\{\beta^{\prime} = \left\lbrack {\frac{1}{2} + {\frac{1}{2^{3}}\left\{ {{\ln \left( {1 - k + {2{kx}^{\prime}}} \right)} - {\ln \left( {1 + k - {2{kx}^{\prime}}} \right)}} \right\}}} \right\rbrack^{n}} & (2)\end{matrix}$

where β′ is normalized camber angle, β_(i) is inlet metal angle, β_(e)is exit metal angle, β is the camber angle at a given normalized chordlocation, x′ is normalized chord, and n and k are constants. In one formk=0.96 and n=0.25.

FIG. 7 depicts the distribution of normalized camber angle againstnormalized chord for one embodiment of the present application plottedagainst distributions for other designs. As will be apparent from thefigure, the depicted embodiment of the present application experiences arelatively high level of turning in the first 5% of true chord and arelatively flat camber-angle distribution for the majority of the chordlength downstream of the leading-edge region. In other embodiments ofthe airflow member 58, the normalized camber angle is at least 0.5 inthe first 10% of normalized chord. In still other embodiments thenormalized camber angle is at least 0.5 in the first 5% of normalizedchord. In yet other embodiments the normalized beta is at least 0.6 inthe first 10% of normalized chord.

In one form the camber angle distribution of the present application canbe expressed as a cumulative integral as shown with the equation

$\begin{matrix}{0.4 < \frac{\int\limits_{0}^{1}{\left\lbrack {\int\limits_{0}^{x}{\beta^{\prime}{x}}} \right\rbrack {x}}}{\int\limits_{0}^{1}{\beta^{\prime}{x}}} \leq 0.5} & (3)\end{matrix}$

where x represents chord. The lower bound of the cumulative integral canrange anywhere between 0.4 and 0.4695. To set forth just a fewnon-limiting examples, in some forms the lower bound of the cumulativeintegral is 0.42. In still other forms the lower bound of the cumulativeintegral is 0.45. In still further forms the lower bound of thecumulative integral is 0.46.

The relationships described above regarding the camber angle are withrespect to a camber angle at a given span location of the airflow member58. In one non-limiting form the camber angle relationships andembodiments described above relate to the camber angle distribution atthe tip of the airflow member 58. In some applications the camber angledistribution described above can be blended into a camber angledistribution of a main part of the airflow member 58. Such a blendingcan start at a variety of span height locations and proceed to the tip.To set forth just one non-limiting example, the blending can start atapproximately ⅔ span of the airflow member 58. Various techniques can beused to blend the main part of the airflow member to the camber angledistribution discussed above. For example, the blending can beaccomplished by a weighted average from a minimum at the blend point toa maximum at the tip. In some embodiments the blending can occur in acubic weighted distribution. Other embodiments can include otherdistributions such as quadratic or hyperbolic tangent distributions, toset forth just two non-limiting alternatives. FIG. 8 discloses anon-limiting weight function of a cubic blending. In the embodimentdisclosed, the streamline number 21 corresponds to full span height. Theblending begins in the embodiment of FIG. 8 at about streamline 15.

The maximum thickness location of the tip camber style can occur at anumber of chord locations, but in one non-limiting embodiment thelocation is about the half true chord location. The location of themaximum thickness can be blended into the main blade maximum thicknesslocation with a weighting distribution following that of the camberblending, to set forth just one non-limiting example.

FIG. 9 illustrates various performance characteristics of one embodimentof the airflow member 58. As can be seen from the figure, greaterefficiencies and pressure ratios are provided across a range of pressureratios and inlet corrected flows.

Some features of the airflow member 58 can be adjusted whenincorporating a tip camber style into a main airflow member 58 style.For example, the trailing edge metal angle may need to be adjusted tocounter the effect of a rapidly varying setting/stagger angle in the tipregion which may be caused by the nature of the camber style at the tipand also the manner in which the tip camber style is blended into themain portion of the airflow member 58.

One aspect of the present application provides an apparatus comprising agas turbine engine having a compressor, a compressor airfoil memberdisposed in the compressor and having a gap between a tip of thecompressor airfoil member and a flow surface of the compressor, thecompressor airfoil member having a camber angle variation along a chordand a concave pressure side surface at the tip of the compressor airfoilmember from a leading edge to an intermediate-chord location thatresults in a normalized beta of at least 0.4 in the first 10% ofnormalized chord, and wherein normalized beta is defined as

$\beta^{\prime} = \frac{\beta_{i} - \beta}{\beta_{i} - \beta_{e}}$

where β_(i) is inlet metal angle, β_(e) is exit metal angle, and β isthe camber angle at a given chord location.

Yet another aspect of the present application provides an apparatuscomprising a gas turbine engine having a compressor, a compressorairfoil member disposed in the compressor and having a camber anglevariation along a chord and a concave pressure side surface at the tipof the compressor airfoil member from a leading edge to anintermediate-chord location that results in a normalized beta of atleast 0.4 in the first 10% of normalized chord, and wherein normalizedbeta is defined as

$\beta^{\prime} = \frac{\beta_{i} - \beta}{\beta_{i} - \beta_{e}}$

where β_(i) is inlet metal angle, β_(e) is exit metal angle, and β isthe camber angle at a given chord location.

One feature of the present application provides wherein the normalizedbeta is at least 0.5 in the first 10% of normalized chord.

Another feature of the present application provides wherein thenormalized beta is at least 0.5 in the first 5% of normalized chord.

A still further feature of the present application provides wherein thenormalized beta is at least 0.6 in the first 10% of normalized chord.

Yet still a further feature of the present application provides hereinthe normalized beta follows the equation

$\beta^{\prime} = \left\lbrack {\frac{1}{2} + {\frac{1}{2^{3}}\left\{ {{\ln \left( {1 - k + {2{kx}^{\prime}}} \right)} - {\ln \left( {1 + k - {2{kx}^{\prime}}} \right)}} \right\}}} \right\rbrack^{n}$

where x′ is normalized chord and n and k are constants.

Yet still another feature of the present application provides whereink=0.96 and n=0.25.

Still yet another feature of the present application provides whereinthe compressor airfoil member is operable to increase a pressure of aworking fluid flowing through the compressor during operation of the gasturbine engine.

Still a further feature of the present application provides means forblending the camber angle variation at the tip of the compressor airfoilmember to a camber angle variation at an intermediate height of thecompressor airfoil member.

Yet another feature of the present application provides wherein thedistribution of normalized beta along the normalized chord includes aninflection point at a mid-chord location.

Another aspect of the present application provides an apparatuscomprising a gas turbine engine including a compressor for increasingthe total pressure of a working fluid prior to delivery to a combustorfor mixing with fuel, the compressor having a plurality of airfoilmembers that extend between a first flow surface and a second flowsurface and form gaps between ends of the airfoil members and the secondwall, the plurality of airfoil members having a camber shape at theirtips that provides a cumulative integral according to the relationship

$0.4 < \frac{\int\limits_{0}^{1}{\left\lbrack {\int\limits_{0}^{x}{B^{\prime}{x}}} \right\rbrack {x}}}{\int\limits_{0}^{1}{B^{\prime}{x}}} \leq 0.5$

where x is chord location,

${\beta^{\prime} = \frac{\beta_{i} - \beta}{\beta_{i} - \beta_{e}}},$

β_(i) is inlet metal angle, β_(e) is exit metal angle, and β is thecamber angle at a given chord location.

Still another aspect of the present application provides an apparatuscomprising a gas turbine engine including a compressor for increasingthe total pressure of a working fluid prior to delivery to a combustorfor mixing with fuel, the compressor having a plurality of airfoilmembers that extend between a first flow surface and a second flowsurface, the plurality of airfoil members having a camber shape at theirtips that provides a cumulative integral according to the relationship

$0.4 < \frac{\int\limits_{0}^{1}{\left\lbrack {\int\limits_{0}^{x}{B^{\prime}{x}}} \right\rbrack {x}}}{\int\limits_{0}^{1}{B^{\prime}{x}}} \leq 0.5$

where x is chord location,

${\beta^{\prime} = \frac{\beta_{i} - \beta}{\beta_{i} - \beta_{e}}},$

β_(i) is inlet metal angle, β_(e) is exit metal angle, and β is thecamber angle at a given chord location.

A feature of the present application provides wherein the lower bound ofthe cumulative integral is 0.42.

Another feature of the present application provides wherein the cambershape at the tips is blended into a main portion of the plurality ofairfoil members at an intermediate blade height.

Still yet another feature of the present application provides whereinthe camber shape at the tips is blended into a main portion of theplurality of airfoil members at about ⅔ of the height of the pluralityof airfoil members.

Still a further feature of the present application provides wherein thecamber shape at the tips is blended into a camber shape in the mainportion of the plurality of airfoil members with a weighted averagebetween the two shapes.

Yet still a further feature of the present application provides whereinthe camber style at the tips is blended into the camber style in themain body of the plurality of blades using a cubic weightingdistribution.

A yet still further feature of the present application provides whereinthe camber shape at the tips includes a maximum thickness-to-chord atabout 50% chord.

Another feature of the present application provides wherein thenormalized beta follows the equation

$\beta^{\prime} = \left\lbrack {\frac{1}{2} + {\frac{1}{2^{3}}\left\{ {{\ln \left( {1 - k + {2{kx}^{\prime}}} \right)} - {\ln \left( {1 + k - {2{kx}^{\prime}}} \right)}} \right\}}} \right\rbrack^{n}$

where x′ is normalized chord, k=0.96, and n=0.25.

Yet another aspect of the present application provides an apparatuscomprising a compressor blade of a gas turbine engine capable of beingrotated to produce a pressure rise in a working fluid during operationof the gas turbine engine, the compressor blade including a main bladeportion having a first camber shape that is blended with a tip portionhaving a second camber shape, the second camber shape producing arelatively low pressure near the leading edge of the tip to induce alocal jet of stream-wise momentum flow.

A feature of the present application provides wherein the second cambershape is blended into the first camber shape at about ⅔ of the height ofthe compressor blade.

Another feature of the present application provides wherein the secondcamber shape is blended into the first camber shape with a weightedaverage.

A further feature of the present application provides wherein themaximum thickness-to-chord of the second camber shape is at about themid-chord location.

A still further feature of the present application provides wherein thesecond camber shape includes a normalized camber angle defined as

$\beta^{\prime} = \frac{\beta_{i} - \beta}{\beta_{i} - \beta_{e}}$

where β_(i) is inlet metal angle, β_(e) is exit metal angle, and β isthe camber angle at a given chord location, and wherein normalizedcamber angle is at least 0.6 in the first 10% of normalized chord.

Still yet another feature of the present application provides whereinthe second camber shape provides a cumulative integral according to therelationship

$0.4 < \frac{\int\limits_{0}^{1}{\left\lbrack {\int\limits_{0}^{x}{B^{\prime}{x}}} \right\rbrack {x}}}{\int\limits_{0}^{1}{B^{\prime}{x}}} \leq 0.5$

where x is chord location,

${\beta^{\prime} = \frac{\beta_{i} - \beta}{\beta_{i} - \beta_{e}}},$

β_(i) is inlet metal angle, β_(e) is exit metal angle, and β is thecamber angle at a given chord location.

Still another feature of the present application provides wherein thelower bound of the cumulative integral is 0.42.

While the invention has been illustrated and described in detail in thedrawings and foregoing description, the same is to be considered asillustrative and not restrictive in character, it being understood thatonly the preferred embodiments have been shown and described and thatall changes and modifications that come within the spirit of theinventions are desired to be protected. It should be understood thatwhile the use of words such as preferable, preferably, preferred or morepreferred utilized in the description above indicate that the feature sodescribed may be more desirable, it nonetheless may not be necessary andembodiments lacking the same may be contemplated as within the scope ofthe invention, the scope being defined by the claims that follow. Inreading the claims, it is intended that when words such as “a,” “an,”“at least one,” or “at least one portion” are used there is no intentionto limit the claim to only one item unless specifically stated to thecontrary in the claim. When the language “at least a portion” and/or “aportion” is used the item can include a portion and/or the entire itemunless specifically stated to the contrary.

We claim:
 1. An apparatus comprising: a gas turbine engine having a compressor; a compressor airfoil member disposed in the compressor and having a camber angle variation along a chord and a concave pressure side surface at a tip portion of the compressor airfoil member from a leading edge to an intermediate-chord location that results in a normalized camber angle that reaches at least 0.4 in the first 10% of normalized chord; and wherein the normalized camber angle is defined as $\beta^{\prime} = \frac{\beta_{i} - \beta}{\beta_{i} - \beta_{e}}$ where β_(i) is inlet metal angle, β_(e) is exit metal angle, and β is a camber angle at a given chord location.
 2. The apparatus of claim 1, wherein the normalized camber angle reaches at least 0.5 in the first 10% of normalized chord.
 3. The apparatus of claim 2, wherein the normalized camber angle reaches at least 0.5 in the first 5% of normalized chord.
 4. The apparatus of claim 1, wherein the normalized camber angle reaches at least 0.6 in the first 10% of normalized chord.
 5. The apparatus of claim 1, wherein the normalized camber angle follows the equation $\beta^{\prime} = \left\lbrack {\frac{1}{2} + {\frac{1}{2^{3}}\left\{ {{\ln \left( {1 - k + {2{kx}^{\prime}}} \right)} - {\ln \left( {1 + k - {2{kx}^{\prime}}} \right)}} \right\}}} \right\rbrack^{n}$ where x′ is normalized chord and n and k are constants.
 6. The apparatus of claim 5, wherein k=0.96 and n=0.25.
 7. The apparatus of claim 1, wherein the compressor airfoil member is operable to increase a pressure of a working fluid flowing through the compressor during operation of the gas turbine engine.
 8. The apparatus of claim 1, wherein the camber angle variation at the tip portion of the compressor airfoil member is blended into a second camber angle variation at an intermediate height of the compressor airfoil member.
 9. The apparatus of claim 1, wherein a distribution of normalized camber angle along a normalized chord includes an inflection point at a mid-chord location.
 10. An apparatus comprising: a gas turbine engine including a compressor for increasing a total pressure of a working fluid prior to delivery to a combustor for mixing with fuel, the compressor having a plurality of airfoil members that extend between a first flow surface and a second flow surface, the plurality of airfoil members having a first camber shape at a tip portion that provides a cumulative integral according to the relationship $0.4 < \frac{\int\limits_{0}^{1}{\left\lbrack {\int\limits_{0}^{x}{B^{\prime}{x}}} \right\rbrack {x}}}{\int\limits_{0}^{1}{B^{\prime}{x}}} \leq 0.5$ where x is chord location, ${\beta^{\prime} = \frac{\beta_{i} - \beta}{\beta_{i} - \beta_{e}}},$ β_(i) is inlet metal angle, β_(e) is exit metal angle, and β is a camber angle at a given chord location.
 11. The apparatus of claim 10, wherein the lower bound of the cumulative integral is 0.42.
 12. The apparatus of claim 10, wherein the first camber shape is blended into a main portion of the plurality of airfoil members at an intermediate blade height.
 13. The apparatus of claim 12, wherein the first camber shape is blended into a second camber shape at the main portion of the plurality of airfoil members at about ⅔ of a height of the plurality of airfoil members.
 14. The apparatus of claim 12, wherein the first camber shape is blended into a second camber shape in the main portion of the plurality of airfoil members with a weighted average between the two shapes.
 15. The apparatus of claim 13, wherein the first camber shape is blended into the second camber shape in the main body of the plurality of blades using a cubic weighting distribution.
 16. The apparatus of claim 10, wherein the first camber shape includes a maximum thickness-to-chord at about 50% chord.
 17. The apparatus of claim 10, wherein the normalized camber angle follows the equation $\beta^{\prime} = \left\lbrack {\frac{1}{2} + {\frac{1}{2^{3}}\left\{ {{\ln \left( {1 - k + {2{kx}^{\prime}}} \right)} - {\ln \left( {1 + k - {2{kx}^{\prime}}} \right)}} \right\}}} \right\rbrack^{n}$ where x′ is normalized chord, k=0.96, and n=0.25. 